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This publication constitutes the refereed lawsuits of the sixth foreign Workshop, IWDW 2007, held in Guangzhou, China, in December 2007. The 24 revised complete papers including three invited papers have been conscientiously reviewed and chosen from eighty one submissions. The papers are prepared in topical sections on watermark defense; steganalysis; authentication; reversible information hiding; strong watermarking; poster consultation; conception and techniques in watermarking.

Due to ¯ = { δ¯1 , δ¯2 . . δ¯n | ∃ i such the preceding deﬁnition, denote the forecasts by F (δ) C C C that δ¯i ≤ 1−β or δ¯i ≤ 1−β }, where 1−β is the break-even point. Suppose that the forecasts for the coming requests are correct, the online investor will buy a Bahncard at the early stage, guaranteeing the competitive ratio rAˆ less than t r∗ . Within the limitation of t r∗ , a risk algorithm Aˆ will compute a restricted optimal ∗ ˆ competitive ratio rA ˆ . The analysis of the risk algorithm A is as follows: ˆ = inf rA ¯ = { δ¯ | Theorem 2.

First, they don’t ask uniformity on the curvature r and second the notion of prox-regularity is deﬁned for a point x ¯ and a subgradient v¯ of the function f. Thus, the class of prox-regular functions is much larger than the r-prox-regular one. 1). In this case we would say that f is r-prox-regular at (¯ x, v¯). For example, for each v¯ (ﬁxed) this notion would be conveniently normalized to the case where x ¯ = 0 and v¯ = 0, moreover with f (0) = 0. By using this terminology we could establish the following lemma, however in the following sections we will always work with the notion of r-prox-regular.

We say that f is r prox-regular at (¯ x, v¯) if and only if g(x) = f (x + x ¯) − v¯, x is r prox-regular at x ¯ = 0, for v¯ = 0. ). In what follows we will study the following problem, namely, ﬁnding critical points x∗ of the problem, minn f (x), (2) x∈IR that is, such that 0 ∈ ∂f (x∗ ), where ∂f (x) is the Mordukhovich subgradient set already called subgradient set in this paper . 2], see Ref. [13]) and which is useful for our purpose. Lemma 2. Suppose that f is r-prox-regular at x ¯, and let λ ∈ (0, 1/r) and g(x) = f (x + x ¯) − v¯, x , where v¯ ∈ ∂f (¯ x).